1 Introduction

The performance of gas turbine engines is evaluated in terms of engine-specific work and efficiency depending on cycle pressure ratio. Modern engines are characterized by higher pressure ratios compared with older ones; consequently, both the size and weight of compressors should be increased. In order to improve the power-to-weight ratio of gas turbine, designers always look for modern techniques to reduce number of compressor stages [1]. This can be achieved by increasing compressor blade loading. Since the flow in the compressor diffuses, compressor aerodynamic losses are affected by the reversed flow associated with an adverse pressure gradient, and the 3D flow separation corresponding to vortex formation through compressor cascades [2,3,4,5]. Compressor cascade aerodynamic losses are caused by the primary (profile) loss, the secondary flow loss, and the tip-leakage loss [6, 7]. The profile losses are formed due to blade boundary layer and wake effects at the blade trailing edge. Secondary flow losses, on the other hand, are claimed to occur owing to presence of horseshoe vortex flow, and crossflow vortex caused by the pressure difference between the pressure and adjacent suction sides [8]. Corner vortices are formed as well, causing corner separation. Vortices can be coupled in 3D patterns to develop a separation zone of low velocity or low momentum flow. The development of the separation zone leads to reduced static pressure rise, flow blockage, and compressor instability. Furthermore, the unsteady nature of the separated flow causes an unfavorable compressor surge, which may cause engine damage [4].

There are several parameters affecting separation structure, such as blade profile, incidence angle, inflow boundary layer, free-stream turbulence, Reynolds number, Mach number, rotating effect, and surface roughness [2]. Many studies in the literature have concluded that an increase in flow incidence increases the likelihood of coroner separation [9,10,11,12,13]. Even though the increase in the blade turning angle is required to improve compressor pressure loading for the same number of stages which is accompanied by an increased separation loss. Thus, it is important to control flow separation. This can be realized using active flow control (AFC) as well as passive flow control (PFC) approaches. In AFC approaches, energy is supplied to the separated flow from a supplementary external source, such as plasma jet actuators [14,15,16], synthetic jet actuators, zero net mass flux actuators [17, 18], blowing [19,20,21], suction [22], suction and blowing [23, 24], and vortex generator jet [25]. However, in PFC approaches, the flow in the separation zone is reenergized by the flow itself without the need for an external energy source. Where a bit of modification is undertaken using a slotted or perforated jet, vortex generators or splitter plates, slats, flaps, trapped vortex, dimples, rumples, and end-wall contouring. The PFC approach is therefore preferred due to its simplicity of structure and implementation, absence of additional moving elements, and economic advantage [26].

The effect of using the PFC method was studied numerically and experimentally to examine its ability to improve compressor cascade performance. Hu et al. [27] studied the effect of making a slot in the blade on compressor flow separation. It was concluded that, for the slotted blade, the suppressed vortices in the flow separation zone associated with the re-energized low momentum flow reduce the loss coefficient by 17.7% for an incident angle of 6° with an insignificant reduction for zero-incident angle. Jiaguo et al. [28] investigated the effect of using a curved slot on highly loaded compressor cascade performance. They managed to reduce the total losses by 21.9%, associated with the ordered outflow that reduces the suction side separation. Hu et al.[29] used the combination of slotted blade and vortex generator flow controllers to obtain further reduction in energy losses (reaching 30.9%) by reducing the size and intensity of the formed vortices. Recently, Hu et al. [30] extended their work to apply this combination in a single-stage transonic axial compressor stator blade and applied only the slotted blade in the rotor. They observed a remarkable improvement in the compressor’s performance and stability with a significant reduction in rotor corner separation with 1.83% and 0.88% improvement in total pressure rise and isentropic efficiency, respectively. Taghavi-Zenouz and Abbasi [31] carried out an unsteady computational study to determine how the grooved rotor shroud of a low-speed axial compressor controls the spike stall. They concluded that this PFC method was able to increase axial speed and decrease air blockage, subsequently improving aerodynamic performance.

Several slot characteristics affect aerodynamic loss, such as width, slope, and location. Ramzi et al. [32] determined the appropriate slot location numerically and discovered it to be at mid-distance between the minimum pressure point and the separation point. In this regard, the slot width should be proportional to the separation size; the larger the separation zone, the wider the required slot [28]. The slot outflow angle is designed to produce an angle between the jet and mainstream flows, as studied by [33, 34]. Liu et al. [35] created an arced slot near the end-wall with a height of 10% span configured in like s-shape. The positive effect of the slot is to minimize separation losses across a wide range of flow incidence angles and with varied geometric cascade designs, as proved numerically and experimentally. Tang et al. [36] used two end-blade slots extended to a 20% span to remarkably improve the pressure loss by providing self-adaptive flow jets according to differential pressure between blade sides. Correspondingly, the corner separation was reduced, and so the blade loading could be improved. Song et al. [37] used the Realizable k-ε model to numerically investigate the effect of passive air injection via hole and slot configurations on the performance of highly loaded compressor cascade based on the NACA 65 profile blade. It was observed that using air injection through a slot reduces the total pressure loss coefficient by 22.15%. Moreover, they used combined 3-hole jets and recorded a further reduction in the total pressure losses, even more than single and double holes. It was concluded that the use of a slotted blade is more effective than using a perforated blade. Zhou et al. [38] discovered experimentally that at low Reynolds numbers ranging from 1.2 × 105 to 2.1 × 105, the slotted blade provided a maximum reduction in wake losses width and depth of 48% and 45%, respectively. Additionally, they used the Spalart–Allmaras turbulence model and the Abu-Ghannam and Shaw transition model. They numerically observed that using a slotted blade effectively reduces the total pressure loss by up to 42% and raises the static pressure by 30% with a 3.2° increment of flow turning angle. Yoon et al. [39] numerically investigated the impact of a slotted rotor blade placed near the end-wall to control the hub corner stall on the performance of an axial compressor. It has been reported that a tiny slot height of less than 10% of the span at the end-wall can raise the stall margin by more than 6% with a little gain in efficiency. The aspirated jet from the pressure side to the suction side eliminates or reduces the origins of corner separation at the hub, resulting in this improvement.

In conclusion, flow separation is a serious phenomenon that negatively affects the efficiency of compressor cascade; therefore, it should be controlled for improving the compressor performance. Considerable research has been conducted to control flow separation either by using AFC or PFC approaches leading to remarkable improvements in the performance and stability of compressors. The current work aims to extend the previous research using numerical and experimental methodologies to investigate passive flow separation control in an arced slotted blade. It also seeks to examine the associated aerodynamics and flow physics besides analyzing the aerodynamic cascade losses. Since accurate separation size prediction is critical for designing effective slot parameters, four turbulence models were tested and evaluated using the experimental results to enhance the prediction of corner separation. Finally, the influence of incidence angle on cascade loss and blade loading was extensively analyzed considering several incidence angles for slotted and un-slotted (original) blades.

2 Experimental setup and measuring instrumentations

The experimental setup consisted of an open-loop low-speed wind tunnel able to provide uniform airflow at the inlet flow velocity of up to 30 m/s and a test section facilitated with necessary instrumentations, as shown in Fig. 1. The air supplied by a centrifugal fan possesses high disturbances, so it was gradually laminated to provide uniform flow velocity through a series of perforated and honeycomb plates. At the end of the wind tunnel, a smooth contraction section was precisely designed to supply uniform air to the test section. The test section contained a linear compressor cascade of six (NACA 65-009) airfoil blades that could be mounted in various wedges, each of which has a specific angle to control the inflow incidence angle. The test section has many measuring ports opened before and after the cascade throughout the medium passage to enable access for measuring flow velocities and pressures via five hole-probe. The measurements were taken at inlet plane A before the blade leading edge by a distance of 1.5 times of axial chord Ca and at plane M downstream the cascade by a distance of 0.091Ca after the blade trailing edge. The geometries of the un-slotted blade with corresponding dimensions and nomenclature are shown in Fig. 2, while Fig. 3 shows the geometry of the slotted blade. Here, the slot is located approximately halfway between the minimum pressure point and the separation point as per the expected location of the separation region on the blade suction side surface [32]. In this study, for an incident angle of \({6}^{^\circ }\), the minimum pressure point, and the separation point were found to exist at 50% and 60% of the blade axial chord; respectively. The inlet and outlet angles of the grooved slot were determined to provide an air jet with high outlet velocity to form a diverged-converged passage in accordance with the literature [28, 34]. Slot width and height were selected to provide sufficient airflow affecting the separation zone and maximize the reduction in pressure loss no matter the tested incidence angles. The measurement system shown in Fig. 4 is based on an L-shape five-hole probe having a tip diameter of 3.18 mm coupled with DAQ (ATX sensor module manufactured by Aero-probe international Company in the USA with a sampling rate of up to \({10}^{4}\) sample per second). The inlet velocity profile in the span-wise direction was measured to use this distribution in the CFD solver, in addition to determining the total pressure loss coefficient (for verifying the proposed numerical simulation procedure). The estimated error in the determined velocity profile per the least square method is about ± 0.3 m/s.

Fig. 1
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Wind tunnel facility with the linear-cascade test section

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Fig. 2
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Cascade nomenclature

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Fig. 3
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Slotted blade design

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Fig. 4
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DAQ with L-shape five-hole pressure probe

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3 Numerical simulations

In this section, the numerical simulation and applied models are briefly described. First, detailed computational domain, grid generation, and boundary conditions are presented. Then, post solver and corresponding data processing are stated. Finally, the turbulence modes used in the present numerical part affecting the prediction of corner separation are provided.

3.1 Grid generation and boundary conditions

The numerical simulation was carried out by ANSYS Fluent solver and the grids for slotted and un-slotted blades were generated using ANSYS ICEM CFD. The computational domain for the original blade provided in Fig. 5 consists of a 12-structured multi-block mesh-like H–O–H topology. While the computational domain with a slotted blade is shown in Fig. 6, where the slot height in the span direction relative to the blade height is selected to be 0.08 from the blade span (\(\cong\) 15 mm height) from the end wall. The grid dependency test was performed using ten grid densities where a good solution consistency was obtained stating from \(3.2\times {10}^{6}\) hexahedral cells up to half span as shown in Fig. 7. The SIMPLEC algorithm and the Quick scheme were applied during data discretization obtained during solution under a pressure-based solver considering steady incompressible air properties. In the current work, four different turbulence models were used and validated by experimental measurements. Enhanced near-wall treatment occurs with \({y }^{+}\approx 0.6\) on average and is used to ensure the accurate flow details in viscous sub-layer. The inlet flow Reynolds number based on the chord length was maintained at value \({R}_{ec}=\frac{\rho {V}_{\infty }C}{\mu }=2.98\times 1{0}^{5}\). The study was carried out considering a wide range of incidence angles from + 6° to − 6°. The inlet velocity distribution was measured to apply at the inlet for the CFD calculations, as shown in Fig. 8. Velocity distribution close to the end-wall could not be measured due to the coarse size of the probe. Therefore, theses data are fitted according to the power law for solving turbulent boundary layer equation as follows:

$$\frac{V}{{V_{\infty } }} = \left( {\frac{Z}{\delta }} \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 7}}\right.\kern-0pt} \!\lower0.7ex\hbox{$7$}}}}$$
(1)

where \(V_{\infty } = 30 \pm 0.3\frac{{\text{m}}}{{\text{s}}},\) δ is the inlet boundary layer thickness δ = 15 mm as seen in Fig. 8, and here z varies from zero to δ. The turbulence intensity is assumed to be 1% and the turbulence length scale \((l)\) is 7% from the hydraulic diameter (\({D}_{H}\)); thus \(l=0.07\times {D}_{H}=17.82 \text{mm}\) [40]. Here, the zero-pressure is used at outlet, at mid-span, a symmetric boundary is applied, and the pitch-wise boundaries are attached by the periodic condition to reduce computational efforts. Both end-wall and blades are assumed to be non-slip.

Fig. 5
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Mesh generation with boundary conditions for the original case

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Fig. 6
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Mesh generation topology for the slotted blade

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Fig. 7
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Grid independence study for TPLC at plane M, \(i = 6^{^\circ }\) using RSM

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Fig. 8
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Velocity profile in span-wise direction at the inlet

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3.2 Data processing

The flow through compressor cascade was numerically simulated according to major cold flow governing equations, including continuity and momentum (Navier–Stokes) equations under steady incompressible flow conditions. The obtained simulation of the flow from the solution of the governing equation in conjunction with the applied turbulence models has been processed to evaluate the compressor performance. The calculated parameters include total pressure loss coefficient \((\omega )\), static pressure coefficient \(({C}_{ps})\), and other loss coefficients. The total pressure losses coefficient is determined by:

$$\omega = \frac{{p_{t\infty } - p_{t} }}{{p_{t\infty } - p_{s\infty } }}$$
(2)

The static pressure coefficient on blade surfaces can described as:

$$C_{ps} = \frac{{p_{s} - p_{s\infty } }}{{p_{t\infty } - p_{s\infty } }}$$
(3)

To evaluate the losses coefficient at a certain plane, the mass-weighted average for the TPLC is calculated from the next equation as:

$$\overline{\omega } = \frac{{\mathop \sum \nolimits_{i = 1}^{n} \omega \rho_{i} \left| {\overrightarrow {{V_{i} }} .\overrightarrow {{A_{i} }} } \right|}}{{\mathop \sum \nolimits_{i = 1}^{n} \rho_{i} \left| {\overrightarrow {{V_{i} }} .\overrightarrow {{A_{i} }} } \right|}}$$
(4)

where \(i,n\) are the cells number, \(V_{i}\) is the average velocity at the cell center, and \(A_{i}\) is the area of the cell. Here, there are two cases of calculation comparison, the original case refers to the un-slotted case and the other with slot. The percentage of losses reduction/increase can be calculated as:

$$\Delta \overline{\omega } = \frac{{\overline{\omega }_{slotted} - \overline{\omega }_{original} }}{{\overline{\omega }_{original} }} \times 100 \,\left[ \% \right]$$
(5)

The average static pressure rise is calculated by the following integration:

$$\overline{{C_{ps} }} = \left. {\mathop \int \limits_{0}^{{C_{a} }} C_{ps} \left( x \right)dx} \right|_{P - S} - \left. {\mathop \int \limits_{0}^{{C_{a} }} C_{ps} \left( x \right)dx} \right|_{S - S}$$
(6)

The percentage of change in static pressure on blade surfaces is evaluated as:

$$\Delta \overline{{C_{ps} }} = \frac{{\left. {\overline{{C_{ps} }} } \right|_{Slotted} - \left. {\overline{{C_{ps} }} } \right|_{original} }}{{\left. {\overline{{C_{ps} }} } \right|_{original} }} \times 100 \left[ \% \right]$$
(7)

3.3 Corner separation turbulence models

To get an accurate solution for flow simulation using a slotted blade (that would be designed later), it is important to select the appropriate turbulence model. For instance, Gibson et al. [41] performed an experimental and computational study using various turbulence models in a centrifugal compressor. They concluded that SA is unable to predict the appropriate flow structures at large separation levels; however, RSM has a better prediction for complex wake and separation flow. In addition, the SST model demonstrated accurate predictions of curved rotational flows in the impeller. Accordingly, many studies used various models for detecting and analyzing the 3D corner separation in axial compressors, as collected in Table 1. Among these turbulence models, only four models were applied in the current study to predict flow separation and to reveal the propagation of the recirculation zone through the tested cascade, including S–A, SST K-ω, R k-ε, and RSM turbulence models.

Table 1 Samples of the usage of turbulence models
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4 Results and discussions

4.1 Effects of selecting turbulence model

The comparison between experimental and numerical results was done at a \({6}^{^\circ }\) incidence angle. Four turbulence models were used (S–A, SST k-ω, R k-ε, and RSM) to select the appropriate turbulence model by validating experimental data. The validation of TPLC contour plot was carried out in the original case at an axial plane M, which was located at (\(\text{X}/{\text{C}}_{\text{a}}\)=1.091%) downstream of the cascade. The vertical and horizontal axes represent the distribution of losses in blade height (span-wise) and pitch-wise directions, respectively, as shown in Fig. 9. For the experimental work, the wake losses are approximately confined between the range of (0.02 ≤ Y/S ≤ 0.15) through the entire span. The higher losses concentrated near the end-wall and the blade suction side approximately extend in span-wise direction in the range of (0.027 ≤ Z/H ≤ 0.18) and in pitch-wise range of (0.05 ≤ Y/S ≤ 0.4) in both directions. Losses decreased and the end-wall boundary layer profile losses filled the remainder range in the pitch-wise direction. The zone of losses was located at the junction corner between the blade suction side and end-wall, which shows the corner separation zone.

Fig. 9
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Effects of turbulence models on TPLC calculations at \(i = 6^{^\circ }\) at plane M

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For the numerical calculations, the four turbulence models ensured the location of corner separation, but with a different size. The RSM turbulence model was in good agreement with the experimental measurements. However, there was little difference in TPLC. In the wake zone, the numerical TPLC had little increase compared to the experimental results. This zone was of a narrow space between the blade trailing edge and the five-hole probe (≅ 10 mm); thus, taking measurements there is challenging. However, the experimental profile losses near end-wall in pitch direction were slightly higher than the numerical values. This was caused because of a blockage effect of the five-hole probe near the end-wall. The mass weighted average of the TPLC for the experimental, S–A, SST k-ω, R k-ε, and RSM were 0.08, 0.107, 0.11, 0.084, and 0.07, respectively. Moreover, the deviation of corner separation zone size was 32.56%, 24.72%, 16.8%, and 4.72% using the SST k-ω, S–A, R k-ε and RSM respectively compared with experimental measurements. The experimental value of TPLC was calculated except for the missing zone near the end-wall and blade suction side. The R k-ε was the second appropriate turbulence model. However, the S–A and SST k-ω turbulence models overestimated the separation zone. These conclusions are compatible with [2, 12, 13, 42, 46].

The reason for this kind of losses is also illustrated in Fig. 10, which shows the wall shear stress lines (Limiting streamlines) that express the propagation of corner separation caused by the interaction of the end-wall and blade suction side boundary layer separation. The separation location changed in both end-wall and blade suction-side. For various turbulence models, the separation lines (SL) started at\(0.1 {\text{C}}_{\text{a}}\),\(0.12 {\text{C}}_{\text{a}}\),\(0.25{\text{ C}}_{\text{a}}\), \(0.35 {\text{C}}_{\text{a}}\) with the percentages of passage vortex blockage of flow area, approximated in pitch-wise direction, were 40%, 44%, 41%, and 40% in S–A,SST k-ω, R k-ε, and RSM turbulence models, respectively. Moreover, in the spanwise direction, the location of the focus point \((\text{F})\) and the attached lines \((\text{AL})\) location changed and started at 0.3,0.27, 0.24, and 0.21 from blade height, respectively.

Fig. 10
figure 10

Effects of turbulence models on limiting streamlines at \(i = 6^{^\circ }\); up End-wall and down blade S–S

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It was observed that the S–A and SST k-ω overpredicted the corner separation zone. However, the R k-ε had sensible prediction of the corner separation zone, while RSM had a good agreement with the experimental measurements of the TPLC as shown in Fig. 11. It was more expensive to find the appropriate turbulence model for the highly separated flow because the slot design was very sensitive to the accurate locations of separation.

Fig. 11
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Experimental and numerical results for TPLC at plane M, and \(i = 6^{^\circ }\)

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4.2 Benefits of separation flow control

The slot was designed according to the size of the vortex generated; as mentioned previously, the slot was designed for the worst case, where the inflow incidence angle was 6° and the largest vortices were found; the slot location depends on the separation line at the end-wall and blade suction side; the slot width depends on the vortex size; and the slope of the slot depends on the angle between the stream velocity and the slot jet velocity. In this study, the geometry of the slot location, width, and slope are constants.

Figure 12 compares the experimental measurements and the numerical predictions of the TPLC contours for the original case; without slot and the slotted blade at plane M \((\text{X}/{\text{C}}_{\text{a}}=1.091)\) downstream the cascade at various inflow incidence angles.

Fig. 12
figure 12

Experimental (Exp.) measurements and CFD calculations of TPLC at plane M (\(\text{X}/{\text{C}}_{\text{a}}=1.091\))

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At i = 6°, for the experimental part, the slot reduced the size of the corner separation and wake losses. For the original case, the maximum losses were concentrated in the pitch-wise direction at 0.08 ≤ Y/S ≥ 0.35. In the direction of the span-wise, it was concentrated up to Z/H = 0.18. However, the slot suppressed the high losses in the pitch-wise direction in the range of 0.1 ≤ Y/S ≥ 0.26. Also, in the spanwise direction up to Z/H = 0.1. Moreover, the slotted blade reduced the boundary layer losses at the end-wall and the wakes in the span-wise direction. For numerical calculations, the slot clearly reduced corner separation size, the boundary layer losses at end-wall, and wakes along blade height. The additional energy from jet velocity mixed with the low velocity in the separation zone that increased the momentum and energy, which suppressed the losses.

At \(i = 4^{^\circ }\), for the experimental part, the slot suppressed the wakes in the blade height direction and corner separation losses, which were concentrated near the blade suction side and end-wall. However, the reduction of the end-wall boundary layer losses was not remarkable. For the numerical part, the slot suppressed the corner losses.

At \(i = 0^{^\circ }\), for experimental part, at original case, the size of maximum losses concentrated at the end-wall in the range of 0.1 ≤ Y/S ≥ 0.12 up to Z/H = 0.1 in the span-wise direction with a small size and wake. However, in the slotted case, these losses were concentrated at the end-wall at 0.1 ≤ Y/S ≥ 0.24 up to Z/H = 0.1 in the blade height direction with increasing the wake losses width. The numerical results agreed with the experimental results in increasing the corner separation losses that were due to the jet velocity, which was small enough to reenergize the low momentum zone.

At \(i = - 2^{^\circ }\), for the experimental part, at original case, the size of the maximum losses; were concentrated at the end-wall in the range of 0.1 ≤ Y/S ≥ 0.12 up to Z/H = 0.05 in the span-wise direction with a small size and wake. However, in the slotted case, these losses were concentrated at the end-wall at 0.1 ≤ Y/S ≥ 0.24 up to Z/H = 0.06 in the blade direction with increasing wakes and the end-wall boundary layer.

It can be concluded that there was a good agreement between the experimental measurements and CFD calculations. The results showed that the size of the corner separation was concentrated at the corner zone of interaction of the end-wall and the blade suction side and increased with increasing the flow incidence angle. As incidence increased, more loading increased the adverse pressure gradient with more reversed flow. The slot was able to reduce the separation losses at the incidence angles of \(6^{^\circ }\) and \(4^{^\circ }\) but the slot slightly increased this loss at the incidence angles \(0^{^\circ }\) and \(- 2^{^\circ }\).

Figure 13 represents the velocity coefficient contours at plane M. The low velocity or low momentum fluid was concentrated near blade suction side and end-wall. The velocity increased in the direction of blade height. The comparison between the experimental measurements and numerical calculations were obtained with varying the inflow incidence angle. A good agreement between the experimental and numerical results was observed. When increasing the incidence angle, the low momentum zone was also increased.

Fig. 13
figure 13

Experimental (Exp.) measurements and CFD calculations of velocity coefficient at plane M (\(\text{X}/{\text{C}}_{\text{a}}=1.091\))

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At \(i=6^{^\circ}\) and \(i = 4^{^\circ }\), the slotted blade increased the velocity in the junction of the end-wall near the blade suction side and improved the passage velocity.

At \(i = 0^{^\circ }\) and \(i = - 2^{^\circ }\), the slot slightly decreased the velocity in the corner zone that was the major reason of increase losses.

Figure 14 confirms the slotted blade’s ability to control separation losses, which were calculated numerically using the mass-weighted average TPLC versus incidence angle. It shows that at \(i = 6^{^\circ } ,4^{^\circ }\) and \(2^{^\circ }\), the percentage of losses reduction was 33%, 28.6%, and 23.3%, respectively. However, the losses slightly increased at \(i = 0^{^\circ } , - 2^{^\circ } , - 4^{^\circ }\), and \(- 6^{^\circ }\) with 4%, 5.5%, 7.1%, and 9.4%, respectively.

Fig. 14
figure 14

CFD mass average for TPLC at plan M (\(\text{X}/{\text{C}}_{\text{a}}=1.091)\)

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The slot location, width, and slope were constant with varying flow incidence. So, the slot was more effective with a large corner separation size until the corner separation reached a certain size where the slotted blade was inefficient to suppress the separation losses. At \(6^{^\circ }\) and \(4^{^\circ }\), the design parameters of the slot were able to suppress losses. However, at \(0^{^\circ }\), and \(- 2^{^\circ }\), the size of the corner separation was small, so, this parameter slightly increased losses.

Another representation that achieves the effectiveness of the slot can be modeled numerically using a comparison of limiting streamlines at the end-wall and blade suction side in both cases of with and without slot, as shown in Fig. 15. The trend of the highest separation losses occurred at the incidence angles \(6^{^\circ } ,4^{^\circ } ,0^{^\circ }\), and \(- 2^{^\circ }\), respectively. The slot was able to reenergize the separation zone, and the focus point (F) strength was reduced and near to behind the blade suction side, and streamlines attached early in case of \(6^{^\circ }\) and \(4^{^\circ }\) incidence angle.

Fig. 15
figure 15

CFD Limiting streamlines for original and slotted cases, up End-wall, and down blade S–S

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At \(i = 6^{^\circ }\), the horseshoe vortex \({(H}_{V})\) did not change in both the original and slotted cases, but the size of the passage vortex \({(P}_{V})\) reduced in the cross-flow, which lead to change of the locations of Focus point (F), separation line \(SL\) and the attached line \((AL)\) to get them close the blade suction side. Also, the wake size was reduced by reducing the trailing edge separation vortex\({(T}_{SV})\). It was observed that a separation line occurred inside the slot but close to the slot walls, which means there was a tiny amount of energy losses across the slot. So, the energy transfer from slot was more efficient to achieve its purpose. In the original case, the \(\text{SL}\) started at \(0.35{\text{C}}_{\text{a}}\) and the passage vortex blocked 40% pitch and the attached lines started at 21% from blade height at blade suction side. But for the slotted case the \(\text{SL}\) was delayed starting at \(0.6{\text{C}}_{\text{a}}\) and the passage vortex blocked 28% pitch with no focus points occurred at the blade suction side. Thus, the slot was able to eliminate the suction side separation except the small separation after the slot exit sharp edge. The slot was designed to cover the starting of separation, that rescued the propagation of reversed flow. That is why the slotted blade was successful.

At \({i=4}^{^\circ }\), the horseshoe vortex \({(H}_{V})\) did not change in both the original and slotted cases, but the size of passage vortex \({(P}_{V})\) reduced in the cross-flow, which lead to change the locations of focus point (F) separation line \((\text{SL})\) and attachment line \((AL)\) to get them close the blade suction side. Also, the wake size reduced by reducing the trailing edge separation vortex \(({T}_{SV})\). It can observe that, for the original case, the \((\text{SL})\) starts at \(0.4{\text{C}}_{\text{a}}\), the passage vortex blocked 30% pitch and the attached lines started at 16.2% from blade height at blade suction side. But in the slotted case, the \((\text{SL})\) was delayed starting at \(0.5{\text{C}}_{\text{a}}\), the passage vortex blocked 22% pitch with no focus points occurred at the blade suction side. Thus, the slot was able to eliminate the suction side separation except the small separation after the slot exit sharp edge.

At incidence angle \(0^{^\circ } , - 2^{^\circ }\), the slot increased separation losses that were clearly obvious in limiting streamlines, which were formed at the end-wall and blade suction side. The slot increased the passage vortex \({(P}_{V})\) and the slotted passage suffered a focus point, which poled flow inside it and formed internal separation and reduced the jet velocity, which lead to reduce the energy transfer through the slot. It can be observed at \(i = 0^{^\circ }\) for the original case that the \(\left( {{\text{SL}}} \right)\) started at \(0.7{\text{C}}_{{\text{a}}}\), the passage vortex blocked 6% pitch and the attached lines started at 8% from blade height at blade suction side. But for the slotted case, the \(({\text{SL}}\)) was started earlier at \(0.65{\text{C}}_{{\text{a}}}\), the passage vortex blocked 15% pitch, and the focus point occurred at 5.4% from the blade span at suction side. At \(i = - 2^{^\circ }\) for the original case, the \(({\text{SL}})\) started at \(0.75{\text{C}}_{{\text{a}}} ,\) the passage vortex blocked 5% pitch and the attached lines started at 6.5% from the blade height at blade suction side. But for the slotted case the \(({\text{SL}})\) is started earlier at \(0.62{\text{C}}_{{\text{a}}}\), the passage vortex blocked 15% pitch, and the focus point occurred at 2.7% from the blade span at the suction side.

In Fig. 16, shows the mass weighted average for TPLC distribution through the span-wise direction for the original and slotted cases with different incidence angles. It is observed that for the original case the losses increased with increasing the flow incidence. The majority of losses originated because of primary and secondary losses. These kinds of losses were concentrated near the end-wall, and it decreased exponentially until reached the center of vortex rings causing a slightly losses increase. Then, losses decayed exponentially until reached a constant value of profile losses because of the wall shear stress and wakes. At incidences \(6^{^\circ } ,4^{^\circ }\) the losses decayed from end-wall to 30%, 27% from blade height, respectively, and 20%, 15% from blade height for the incidences \(0^{^\circ } , - 2^{^\circ }\), respectively. At \(i = 6^{^\circ }\), the slot was able to reduce loss coefficient up to 30% span. After this, up to mid span, a tiny amount of reduction was obtained. At \(i = 4^{^\circ }\), the slot was able to reduce losses coefficient up to 27% span. After this, up to mid span, approximately no reduction was found. At \(i = 0^{^\circ }\), the slot increased losses up to 12% span. After this, a slight loss increased up to midspan. At \(i = - 2^{^\circ }\), the slot increased losses up to 13% span. After this, approximately no change of losses occurred up to midspan.

Fig. 16
figure 16

Losses distribution in span-wise direction for original case (solid lines) and slotted blade (dashed lines)

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The low momentum zone is located near the end-wall and blade suction side, and it forms the corner separation. The TPLC contours are plotted in a span-wise direction near end-wall for various incidence angles at a plane located at \((\text{Z}/\text{H}=0.027)\) which a highly losses concentrated on it to show the propagation of losses in pitch -wise direction as shown in Fig. 17, the results show that the high losses are found at the blade suction side and it decreases in pitch-wise direction. At incidences \({6}^{^\circ },{4}^{^\circ }\) the slot is more effective to suppress this loss because of the slot position which is just a separation zone, however at incidences \({0}^{^\circ },{-2}^{^\circ }\) this little loss was originally formed after the separation position and the slot jet slope far away from its main target and the slotted blade was unable to suppress losses moreover, the interior region of the slot suffered from high losses making a low energy transfer to achieve its purpose. The mechanism of action also illustrated using the normalized velocity profiles at the blade suction side at the plane \(\text{Z}/\text{H}=0.027\) as shown in Fig. 18, at \(i={6}^{^\circ }\) the original reversed flow starts approximately at \(\text{X}/{\text{C}}_{\text{a}}=0.5\) and increases the boundary layer separation under the action of adverse pressure gradient, the outlet slot jet flow mixed with the reversed velocity with a higher velocity to prevent the occurrence of reversed flow and remove separation so, the source of small losses generated by the effects of the boundary layer, at \(i={-2}^{^\circ }\) the original reversed flow starts approximately at \(\text{X}/{\text{C}}_{\text{a}}=0.88\) with a little reversed flow zone faring from the slot location making the outlet jet flow effective for the adjacent non-reversed boundary layer and slightly delaying the separation at \(\text{X}/{\text{C}}_{\text{a}}=0.9\),but increasing the size of reversed flow zone in this boundary layer so the losses increase.

Fig. 17
figure 17

CFD calculations of TPLC contours on span-wise planes at \((\text{Z}/\text{H}=0.027)\)

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Fig. 18
figure 18

CFD; 2-D normalized velocity profile on blade S–S for original and slotted cases at \((\text{Z}/\text{H}=0.027)\)

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Figure 19 shows that the original blade static pressure coefficient was affected by the incidence angle plotted at mid-span at which the flow regarded as 2-D flow. And the main source of losses was due to the profile losses. This coefficient increased with the increase of the flow incidence angle up to the stall angle where it decreased. The phenomenon besides the leading edge is the occurrence of a stagnation point. According to the understanding of the term, the stagnation point was positioned at the point where the static pressure achieves a maximum. The value of \({C}_{ps}\) at the stagnation point is roughly 1.0 at mid-span.

Fig. 19
figure 19

CFD static pressure coefficient distribution on original blade surfaces at mid-span

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At negative incidences, the stagnation point was located on the suction side. Whereas it was located on the pressure side at \(0^{^\circ }\) and other positive incidences. At the same time, the stagnation point moved downstream when incidence increased from \(0^{^\circ }\) to \(6^{^\circ }\) as shown in enlarge Fig. 19.

Also, the phenomenon along the trailing edge was explored, as shown in enlarged Fig. 19. The static pressure on the pressure side decreased and then increased very near to the trailing edge, 0.99 ≤ X/Ca ≤ 1.

It was observed that for incidences \(- 4^{^\circ } {\text{and}} - 6^{^\circ } { }\) up to approximately 3% and 7.5% from axial chord, respectively, the pressure distribution at suction side was higher than the pressure side, leading to suppressing the blade loading. The pressure difference between (P-S) and suction side (S–S) decreased with increasing the blade chord. As increasing the pressure difference between the blade pressure side and suction side, the blade loading increased which represented the amount of added work to the flow. The advantage of flow control was to increase blade loading without expending any additional compressor power input or keeping a certain value with lowering stage numbers.

The stagnation \({C}_{ps}<1\) was near the end-wall This was mainly caused by the boundary layer. Therefore, the dynamic pressure in the region near the end-wall was smaller than that at mid-span. The stagnation points near the end-wall also moved downstream with the increasing incidence. This means that the incidence increased from the end-wall to the mid-span, mainly due to the influence of inlet boundary layer and the blockage of the corner stall [46].

Figure 20 shows an average value of static pressure coefficient for the original blade with different incidence angles. When increasing the flow incidence, the blade loading also increases until reaches stall angle, where it decreases, here the second order curve fitting equation is:

$$\overline{{C_{ps} }} = - 0.0018 i^{2} + 0.0303 i + 0.3844$$
(8)
Fig. 20
figure 20

Average value of static pressure coefficient with incidence for original cases

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This fitting was selected to be second order with a root mean square error equal to 99.65% that is the reason of selecting second order rather than other high order equations. From mathematical analysis for this equation, the incidence angle, which maximized the static pressure coefficient was approximately \({8.4}^{^\circ }\).

Figure 21 shows the contours of the static pressure coefficient on blade pressure side and suction side in cases of the slot and without the slot. On the pressure side near the leading edge, the static pressure near the end-wall was smaller than that far from the end-wall. In contrast, on the suction side near the leading edge, the static pressure near the end-wall was larger than that far from the end-wall. This was due to the existence of inlet flow boundary layer. The oblique of the contours close to the end-wall on both the pressure and suction sides were caused by the blockage of corner separation. At incidences \({6}^{^\circ }\), the slot increased the static pressure distribution at blade suction side and pressure side just after the slot location, which increased the blade loading. For the original case, \({C}_{ps}\) at blade suction side near the leading edge, had a negative value, which increased as it far away from leading edge. In addition, at the blade pressure side, where the stagnation point was located, \({C}_{ps}\) increased near the leading edge and decreased going away from the leading edge. Moreover, the \({C}_{ps}\) near the end-wall was small and increased moving away from the end-wall up to the mid-span owing to the effects of end-wall boundary layer separation. At incidences \({-2}^{^\circ }\). For the original case, the \({C}_{ps}\) at the blade suction side near the leading edge was a negative value and increased as it was far away from leading edge. In addition, at the blade pressure side, where the stagnation point was located, \({C}_{ps}\) increased near the leading edge and decreased as moving from the leading edge. Moreover, \({C}_{ps}\) near the end-wall was small and increased moving away from the end-wall up to the mid-span. That happened because of end-wall boundary layer separation. The slotted blade was insignificantly decreasing the \({C}_{ps}\) distribution for both blade suction and pressure surfaces. Moreover, at the surrounding zone of slot, \({C}_{ps}\) decreased because of the separation losses.

Fig. 21
figure 21

Static pressure coefficient distribution at the blade surfaces, left (original) and right (slotted) at a \(i = 6^{^\circ }\) and b \(i = - 2^{^\circ }\)

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Figure 22 represents the effects of the slotted blade jet on the static pressure coefficient at incidence angles \(6^{^\circ }\) and \(- 2^{^\circ }\). The high losses were concentrated near the end-wall, which were reduced with increasing blade span. At \(i = 6^{^\circ }\), the slotted blade improved the blade loading by increasing the pressure coefficient by 17.6%, 17%, and 14.3% at 0.027, 0.25, and 0.5 from blade height, respectively. However, the slotted blade reduced the pressure coefficient by 11%, 2%, and 1.86% at 0.027, 0.25, and 0.5 of the blade heights, respectively, as collected in Fig. 23.

Fig. 22
figure 22

CFD Static pressure coefficient distribution at different blade heights a left at \(i={6}^{^\circ }\) and b right at \(i = - 2^{^\circ }\)

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Fig. 23
figure 23

Average static pressure coefficient for the un-slotted and slotted blade

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5 Conclusions

In the present paper, linear-cascade flow separation was alleviated using the arced diverging-converging slot, which passed from blade P-S to blade S–S with a height of 8% of the blade span near the end-wall. This investigation was performed experimentally and numerically. Numerically, four turbulence models (S–A, R k-ε, SST k-ω and RSM) were tested and experimentally verified. The following conclusions can be drawn:

  1. 1.

    The results of flow separation topology are sensitive to the turbulence model chosen.

  2. 2.

    The deviation of corner separation zone size was 32.56%, 24.72%, 16.8%, and 4.72% using the SST k-ω,S–A, R k-ε, and RSM, respectively, compared with experimental measurements. As a result, the RSM turbulence model had a very good agreement with the experimental data of the corner separation, and the R k-ε model had a sensible prediction of the corner separation. However, the S–A and SST k-ω models overpredicted the corner separation.

  3. 3.

    The separation structure was investigated at various flow incidence angles, concluding that the corner separation increased with increasing the incidence angle.

  4. 4.

    The passive separation control using a slotted blade has a great effect on the separation reduction and improves the compressor cascade performance.

  5. 5.

    Passive techniques can reduce separation losses, which are represented in the TPLC. At incidences \(6^{^\circ }\), \(4^{^\circ }\) and \(2^{^\circ }\), the loss coefficient was reduced by 33%, 28.6%, and 23.3%, respectively. However, the loss coefficient at \(0^{^\circ }\), \(- 2^{^\circ }\), \(- 4^{^\circ }\), and \(- 6^{^\circ }\) was slightly increased by 4%, 5.5%, 7.1%, and 9.4%, respectively.

  6. 6.

    The blade loading increased with the incidence angle up to the stall angle. The slotted blade was able to increase the pressure coefficient by 17.6%, 17%, and 14.3% at \(6^{^\circ }\) incidence. However, the pressure coefficient was decreased by 11%, 2% and 1.86% at \(- 2^{^\circ }\) incidence at planes located 0.027, 0.25 and 0.5 of blade height, respectively.